Theorem int0 4921

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4453	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4913	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 234	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 267	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2757	1 ⊢ ∩ ∅ = V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  ∅c0 4278  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-dif 3901  df-nul 4279  df-int 4907

This theorem is used by:  unissint  4931  uniintsn  4944  rint0  4947  intex  5304  intnex  5305  oev2  8509  fiint  9296  elfi2  9384  fi0  9390  cardmin2  10052  00lsp  21218  cmpfi  23688  ptbasfi  23862  fbssint  24119  fclscmp  24311  zarcmplem  34447  rankeq1o  36854  bj-0int  37942  heibor1lem  38663  ipoglb0  50024  mreclat  50027